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Voxels near tissue borders in medical images contain useful clinical information, but are subject to severe partial volume (PV) effect, which is a major cause of imprecision in quantitative volumetric and texture analysis. in robustness, consistency and quantitative precision were noticed. Results from both synthetic digital phantoms and real patient bladder magnetic resonance images were presented, demonstrating the accuracy and efficiency of the presented theoretical MAP-EM solution. [7] proposed a PV image segmentation algorithm, as an improvement, that directly estimated the tissue components in each voxel via down-sampling. Theoretically, this discrete down-sampling algorithm approaches to a RAC1 continuous solution after infinite numbers of down-sampling operations, which in practice, however, is not achievable. Toward that end, directly modeling PV effect as tissue mixture fractions inside buy 638-94-8 each buy 638-94-8 voxel, in a continuous space, is desirable. In our previous work [8C9], a PV segmentation approach utilizing expectation-maximization (EM) algorithm has been explored trying to simultaneously estimate (1) tissue mixture fractions inside each voxel and (2) statistical model parameters of the image data under the principle of maximum (MAP). As such, PV effect was modeled in a continuous mixture space under the constraint of no more than two tissue types [8] present in each voxel. Moreover, approximated MAP-EM solutions were taken in exchange for less computational complexity [9]. In this paper, we endeavored to acquire the theoretical MAP-EM solution for more general cases, such that the number of tissues considered for each voxel was extended to three and four. Instead of our approximated MAP-EM solution in a quadratic format, the closed-form theoretical solution existed in a set of nonlinear equations up to fifth-order, where numerical equation-solving methods, like Newton and QR, were employed in our study. Via quantitative performance analysis on the synthetic images, the derived exact solution displayed advantages in (1) consistency between two consecutive EM iterations and (2) robustness to data overflow. The remainder of this paper is organized as follows. Section II firstly reviewed the rationale behind MAP-EM algorithm by introducing statistics-based mixture models. Section III fully tabulated the theoretical solution to MAP-EM estimation by discussing different mixture cases, followed by Section IV and V where synthetic computer simulations and patient magnetic resonance imaging (MRI) bladder data were conducted respectively for qualitative evaluation purpose. Finally conclusions were drawn in Section VI. II. MATERIALS AND METHODS In this section, MAP-EM segmentation algorithm addressing PV effect is briefly reviewed, based on the assumption that each voxel contains up to tissue types, each of which shared a certain mixture fraction. A. Review of MAP-EM Segmentation A.1. Image Data Model It is assumed that the acquired image Y is represented by a column vector into the form of {= 1, , denotes the total number of voxels in the image, where subscript indexes voxel of interest and is an observation of current voxel with mean and variance , i.e., = 1,, then given statistical means and variances of {= 1, , = 1,tissue types simultaneously occurring inside each voxel, where the contribution of tissue type to observation is denoted by = 1,,= 1,,is also a random variable having mean and variance , i.e., = 1, , = 1,, by considering the contribution of is depicted as follows, is assumed to be the mixture fraction of tissue type inside subject to and 0 1 , and by defining and as the mean and variance of tissue type fully filling in voxel , we have is considered as an incomplete observation, while the underlying contribution of each tissue type , is complete while invisible, related buy 638-94-8 to via the following conditioning integral, expectation-maximization (MAP-EM) by introducing a Markov Random Field (MRF) penalty term to define an distribution for tissue mixture fraction around its neighbors, such that ML-EM becomes MAP-EM. Applying a Gibbs model on the MRF framework, the penalty on has the general form buy 638-94-8 of are the surrounding neighbors around , is a normalization constant and is an adjustable parameter controlling the degree of the penalty. The exponential energy function (.) can be written as a quadratic form like is a weighing factor for regularizing different orders of neighbors. B. Theoretical Solutions to MAP-EM PV Segmentation In this section, the theoretical solutions to MAP-EM PV segmentation are given by discussing different tissue mixture cases, = 2 and = 3 associated with each voxel, although the total number of tissue types inside the body can be far beyond = 2 and = 3 respectively. B.1. Theoretical Solutions for K =.